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Distributed delays in a hybrid model of tumor-Immune system interplay
Approximate smooth solutions of a mathematical model for the activation and clonal expansion of T cells
1. | Department of Mathematics and Informatics, University of Messina, Viale F. Stagno d'Alcontres n.31, 98166 Messina, Italy, Italy |
2. | Department I.C.I.E.A.M.A., University of Messina, Contrada Di Dio (S.Agata), 98166 Messina, Italy |
References:
[1] |
M. Dolfin and D. Criaco, A phenomenological approach to the dynamics of activation and clonal expansion of T cells,, Mathematical and Computer Modelling, 53 (2011), 314.
|
[2] |
G. Boillat, "La Propagation des Ondes,", $1^{st}$ edition, (1965).
|
[3] |
G. Boillat, Ondes asymptotiques nonlineaires,, Annali di Matematica Pura ed Applicata, IV, CXI (1976), 31.
|
[4] |
D. Fusco, Onde non lineari dispersive e dissipative,, Bollettino U.M.I, 16-A (1976), 450.
|
[5] |
A. Jeffrey and T. Taniuti, "Nonlinear Wave Propagation,", $1^{st}$ edition, (1964).
|
[6] |
A. Jeffrey, The propagation of weak discontinuities in quasilinear symmetric hyperbolic system,, Z. A. M. P. 14 (1963), 14 (1963), 31.
|
[7] |
A. Jeffrey and T. Kakutani, Weak nonlinear dispersive waves: A discussion centered around the Korteweg-de Vries equation,, SIAM Review, 14 (1972), 582.
|
[8] |
A. Jeffrey, The development of jump discontinuities in nonlinear hyperbolic systems of equations in two independent variables,, Arch. Rational Mech. Anal., 14 (1963), 27.
|
[9] |
P. D. Lax, Nonlinear hyperbolic equations,, Comm. Pure Appl. Math., 6 (1983), 231.
|
[10] |
A. Georgescu, "Asymptotic Treatment of Differential Equations,", $1^{st}$ edition, (1995).
|
[11] |
A. Donato and A. M. Greco, "Metodi Qualitativi per Onde Non Lineari - Quaderni del C. N. R., Gruppo Nazionale di Fisica Matematica, 11th Scuola Estiva di Fisica Matematica, Ravello, (1986), 8-20 September,", $1^{st}$ edition, (1987). Google Scholar |
[12] |
Y. Choquet-Bruhat, Ondes asymptotiques et approchees pour systemes d'equations aux derivees partielles nonlineaires,, J. Math. Pures et Appl., 48 (1968), 117.
|
[13] |
P. D. Lax, "Contributions to the Theory of Partial Differential Equations,", $1^{st}$ edition, (1954).
|
[14] |
P. D. Lax, Hyperbolic systems of conservation law (II),, Comm. Pure Appl. Math., 10 (1957), 537.
doi: 10.1002/cpa.3160100406. |
[15] |
T. Taniuti and C. C. Wei, Reductive pertubation method in nonlinear wave propagation,, J. Phys. Soc. Japan, 24 (1968), 941. Google Scholar |
[16] |
V. Ciancio and L. Restuccia, Nonlinear dissipative waves in viscoanelastic media,, Physica A, 132 (1985), 606.
|
[17] |
V. Ciancio and L. Restuccia, Asymptotic waves in anelastic media without memory (Maxwell media),, Physica A, 131 (1985), 251.
doi: 10.1016/0378-4371(85)90090-1. |
[18] |
V. Ciancio and L. Restuccia, The generalized Burgers equation in viscoanelastic media with memory,, Physica A, 142 (1987), 309. Google Scholar |
[19] |
A. Jeffrey, "Quasilinear Hyperbolic Systems and Waves,", $1^{st}$ edition, (1976).
|
[20] |
I. Muller, "Thermodynamics,", $1^{st}$ edition, (1985). Google Scholar |
[21] |
I. Muller and T. Ruggeri, "Rational Extended Thermodynamics,", $1^{st}$ edition, (1998).
|
[22] |
R. M. Ford and D. A. Lauffenburger, Analysis of chemotactic bacterial distributions in population migraton assays using a mathematical model applicable to steep ar shallow attractant gradients,, Bullettin of Mathematical Biology, 53 (1991), 721. Google Scholar |
[23] |
D. A. Lauffenburger and K. H. Keller, A Effects of leukocyte random motility and chemotaxis in tissue inflammatory response,, Theoretical Biology, 81 (): 475. Google Scholar |
[24] |
A. Tosin, D. Ambrosi and L. Preziosi, Mechanics and chemotaxis in the morphogenesis of vascular networks,, Mathematical Biology, 68 (2006), 1819.
doi: 10.1007/s11538-006-9071-2. |
[25] |
H. Byrne and L. Preziosi, Modelling solid tumour growth using the theory of mixtures,, Mathematical Medicine and Biology, 20 (2003), 341.
doi: 10.1093/imammb/20.4.341. |
[26] |
G. Carini, "Lezioni di Istituzioni di Fisica Matematica,", edition, (1989). Google Scholar |
[27] |
J. D. Murray, "Mathematical Biology, vol I,", $2^{nd}$ edition, (2002).
|
[28] |
J. D. Murray, "Mathematical Biology, vol II,", $2^{nd}$ edition, (2002).
|
[29] |
E. Hopf, The partial differential equation ut + uux = xx,, Comm. Appl. Math., 3 (1950), 201.
|
show all references
References:
[1] |
M. Dolfin and D. Criaco, A phenomenological approach to the dynamics of activation and clonal expansion of T cells,, Mathematical and Computer Modelling, 53 (2011), 314.
|
[2] |
G. Boillat, "La Propagation des Ondes,", $1^{st}$ edition, (1965).
|
[3] |
G. Boillat, Ondes asymptotiques nonlineaires,, Annali di Matematica Pura ed Applicata, IV, CXI (1976), 31.
|
[4] |
D. Fusco, Onde non lineari dispersive e dissipative,, Bollettino U.M.I, 16-A (1976), 450.
|
[5] |
A. Jeffrey and T. Taniuti, "Nonlinear Wave Propagation,", $1^{st}$ edition, (1964).
|
[6] |
A. Jeffrey, The propagation of weak discontinuities in quasilinear symmetric hyperbolic system,, Z. A. M. P. 14 (1963), 14 (1963), 31.
|
[7] |
A. Jeffrey and T. Kakutani, Weak nonlinear dispersive waves: A discussion centered around the Korteweg-de Vries equation,, SIAM Review, 14 (1972), 582.
|
[8] |
A. Jeffrey, The development of jump discontinuities in nonlinear hyperbolic systems of equations in two independent variables,, Arch. Rational Mech. Anal., 14 (1963), 27.
|
[9] |
P. D. Lax, Nonlinear hyperbolic equations,, Comm. Pure Appl. Math., 6 (1983), 231.
|
[10] |
A. Georgescu, "Asymptotic Treatment of Differential Equations,", $1^{st}$ edition, (1995).
|
[11] |
A. Donato and A. M. Greco, "Metodi Qualitativi per Onde Non Lineari - Quaderni del C. N. R., Gruppo Nazionale di Fisica Matematica, 11th Scuola Estiva di Fisica Matematica, Ravello, (1986), 8-20 September,", $1^{st}$ edition, (1987). Google Scholar |
[12] |
Y. Choquet-Bruhat, Ondes asymptotiques et approchees pour systemes d'equations aux derivees partielles nonlineaires,, J. Math. Pures et Appl., 48 (1968), 117.
|
[13] |
P. D. Lax, "Contributions to the Theory of Partial Differential Equations,", $1^{st}$ edition, (1954).
|
[14] |
P. D. Lax, Hyperbolic systems of conservation law (II),, Comm. Pure Appl. Math., 10 (1957), 537.
doi: 10.1002/cpa.3160100406. |
[15] |
T. Taniuti and C. C. Wei, Reductive pertubation method in nonlinear wave propagation,, J. Phys. Soc. Japan, 24 (1968), 941. Google Scholar |
[16] |
V. Ciancio and L. Restuccia, Nonlinear dissipative waves in viscoanelastic media,, Physica A, 132 (1985), 606.
|
[17] |
V. Ciancio and L. Restuccia, Asymptotic waves in anelastic media without memory (Maxwell media),, Physica A, 131 (1985), 251.
doi: 10.1016/0378-4371(85)90090-1. |
[18] |
V. Ciancio and L. Restuccia, The generalized Burgers equation in viscoanelastic media with memory,, Physica A, 142 (1987), 309. Google Scholar |
[19] |
A. Jeffrey, "Quasilinear Hyperbolic Systems and Waves,", $1^{st}$ edition, (1976).
|
[20] |
I. Muller, "Thermodynamics,", $1^{st}$ edition, (1985). Google Scholar |
[21] |
I. Muller and T. Ruggeri, "Rational Extended Thermodynamics,", $1^{st}$ edition, (1998).
|
[22] |
R. M. Ford and D. A. Lauffenburger, Analysis of chemotactic bacterial distributions in population migraton assays using a mathematical model applicable to steep ar shallow attractant gradients,, Bullettin of Mathematical Biology, 53 (1991), 721. Google Scholar |
[23] |
D. A. Lauffenburger and K. H. Keller, A Effects of leukocyte random motility and chemotaxis in tissue inflammatory response,, Theoretical Biology, 81 (): 475. Google Scholar |
[24] |
A. Tosin, D. Ambrosi and L. Preziosi, Mechanics and chemotaxis in the morphogenesis of vascular networks,, Mathematical Biology, 68 (2006), 1819.
doi: 10.1007/s11538-006-9071-2. |
[25] |
H. Byrne and L. Preziosi, Modelling solid tumour growth using the theory of mixtures,, Mathematical Medicine and Biology, 20 (2003), 341.
doi: 10.1093/imammb/20.4.341. |
[26] |
G. Carini, "Lezioni di Istituzioni di Fisica Matematica,", edition, (1989). Google Scholar |
[27] |
J. D. Murray, "Mathematical Biology, vol I,", $2^{nd}$ edition, (2002).
|
[28] |
J. D. Murray, "Mathematical Biology, vol II,", $2^{nd}$ edition, (2002).
|
[29] |
E. Hopf, The partial differential equation ut + uux = xx,, Comm. Appl. Math., 3 (1950), 201.
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